Question

If $a^x=b^y=c^z$ and $b^2=ac$, prove that $y=\frac{2xz}{x+z}$.

Answer 1

Given,

$a^x=b^y=c^z$

$\Rightarrow a=k^{\frac{1}{x}},b=k^{\frac{1}{y}},c=k^{\frac{1}{z}}$

$b^2=ac$

$(k^{\frac{1}{y}}{)}^2=(k^{\frac{1}{x}}\left)\right(k^{\frac{1}{z}})$

$k^{\frac{2}{y}}=k^{\frac{1}{x}+\frac{1}{z}}$

$\frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

$\therefore \frac{2}{y}=\frac{x+z}{xz}$

$y=\frac{2xz}{x+z}$